Quart. J. R . Met. Soc. (1975), 101, pp. 529-536 551.553.1 1

On the asymmetry of the land-breeze sea-breeze circulation

5y R. A. PEARSON

Wathematics Department, Monash University, Clayton. Victoria, 31168, Australia

(Received 4 April 1974: revised 6 December 1974. Communicated by Dr. R. H . Clarke)

SUMMARY

Land- and sea-breeze circulations are simulated in a mathematical model, using a stably stratified atmosphere as a basic state. The results show that the land-breeze is weaker than the sea-breezeeven although the heat contrast at the coast is identical in both cases. This is consistent with estimates of the corresponding model energy available potential.

1. INTRODUCTION

It is observed that the land-breeze is usually weaker than the sea-breeze. The different intensity is explained by different heat transfers (Defant 1951); for the land-breeze the night-time stable layer inhibits heat transfer, while during the day convection transfers the heat farther up into the air.

Although the sea-breeze has been modelled numerous times (Pearce 1955; Estoque 1960; McPherson 1970), the land-breeze has received little attention. Neumann and Mahrer (1971) numerically integrate their model over a number of land-sea-breeze cycles. The boundary layer formulation which they used included differing mechanisms for heat transfer in stable and unstable conditions (Estoque 1962). This results in differing heat transfer characteristics (Fig. I ) and also a weaker land-breeze.

In this paper a model used previously (Pearson 1973) to study the sea-breeze is used to study further the differences between the land- and the sea-breeze.

- 4Krn.lNLAND ----- 20Km INLAND

Figure 1. Heat transfer to the ground, from Neumann and Mahrer (1971).

2. THE MODEL

This is a two-dimensional model and includes rotation with the hydrostatic approxima- tion

D u p t - j v = -aptlax (1)

DvlDt + fu = 0 . (2) 529

530 R. A. PEARSON

h / D t - N’w = g,A(t)B(x) ( 1 - :3; - z < z r . = o z > Zt

(4)

aqax + aw/az = 0. ( 5 )

Here o = -gO’/Oo, N 2 = -(g/Oo) (dOo/dz), p f = p/po as the Boussinesq form of the equations has been used. These equations and the basic model have been discussed in Pearson (1973). The motion is forced by the heat input, the right-hand side of Eq. (4). This forcing over the land alone is assumed to be separable with linear height dependence; o, is positive for cooling and negative for heating.

The form for A ( t ) and B(x) are those used in Pearson (1973):

t < O

where a = (2/n) arctan (t,/e). t , is an arbitrary time at which the heating or cooling is assumed to cease.

f 0; x < -x, ... arctan(x/xd) + arctan(x,/xd)

2 arctan(x,/xd) ; - X , Q X B X , B(x) =

x > x,

A stream function $ is now introduced and the vorticity equation derived. A second order quadratically conservative finite difference scheme is used to integrate these equations (see Pearson 1973). The top of the model is a rigid lid, while at the sides the dominant internal gravity waves are not reflected. All other waves will be partially reflected (Pearson 1974).

3. THJ3 AVAILABLE POTENTIAL ENERGY

The total energy in a closed system is the sum of the kinetic and potential energies. The total potential energy can be separated into available potential energy and unavailable potential energy (Lorenz 1967). This available potential energy is an upper bound for the energy that can be converted into kinetic energy by adiabatic processes. In meteorology available potential energy calculations have been used mainly in studies of the general circulation of the atmosphere, and Lorenz shows that in the atmosphere the available potential energy is given approximately by

A.P.E. = fc,rDFZ{(rD - r)q-l where r D = dry adiabatic lapse rate

r = observed lapse rate T = horizontal mean temperature T’ = temperature deviation from the mean.

In his derivation Lorenz has used the hydrostatic approximation and T‘/T = Of/O. If this latter approximation is not used it can be shown that the available potential energy is given to the same approximation by -

A.P.E. = 0 * ’ ~ / 2 N * ~ ( z , t ) (6)

THE LAND-BREEZE SEA-BREEZE CIRCULATION 53 1

where the variables N*(z,t) and a*(z,t) are defined by

Thus N*2(z,t) is defined at any instant by the mean sounding. When the fluid is being heated, the mean sounding will be a function of time. a*'(z,t) is defined as the difference from the mean sounding of the negative buoyancy. This expression for the A.P.E., (6) , is equivalent to that used by Green (1970). We shall use it to calculate the A.P.E. averaged over the whole volume of the model, assuming that it is independent of the rest of the atmosphere.

4. RESULTS AND DISCUSSION

The model atmosphere is assumed to be initially at rest. The sea- or land-breeze is then simulated by heating or cooling the air over the land and numerically integrating the equations. For the results discussed below, the model parameters are (a81 = 12.5cm s-', z, = 1.25km, t,,, = 2h, E = 5h, x,,, = 30km, x, = 15km.

TIME 6 600 HOURS

- 165.0 1650

-1650 165.0

Figure 2. Comparison between the stream function fields for the seaibreeze (above) and the land-breeze (below). The initial heated (or cooled) air is shown by the cross-hatched region in the upper figure. All contours are scaled by 1 x 10'-4(cm2s-1) with a contour interval, above of 5 x 1O6(crn2s-') and

below of 2 x 106(cm2s-').

With these parameters the height at which cooling of the air ceases for the land-breeze is much larger than that observed. However, this allows the vertical distribution of the temperature contrast in the air to be the same in both these model runs. As the magnitude of the temperature contrast at the ground is the same, the total heat gained or lost by the air is identical for both the land- and sea-breeze circulations modelled here.

When the model results are examined, differences in the flow (Fig. 2) and temperature (Fig. 3) field can be observed. The velocity change on the arrival of the sea-breeze is

532

- 303 302

4 301

R. A. PEARSON

TIME IS 6.00 HOURS

,125 -162.5 162.5

Figure 3. The temperature fields after six hours for the sea-breeze (above) and land-breeze (below). All temperatures in K with a contour internal of 1K.

sharper (Fig. 5). The velocity of the sea-breeze is larger than the velocity of the land-breeze (Figs. 4 and 5). More air is transported across the coast by the sea-breeze (Fig. 6). The kinetic energy of the sea-breeze inside the model domain is also larger (Fig. 7). Thus, overall, the sea-breeze is stronger than the land-breeze. These differences occur even although the total heat transfer is identical in both cases.

One would expect the difference to be associated with differences in the available potential energy with the land cooled or heated. For simplicity the A.P.E. is calculated using Eq. (6). The heat averaged over the volume of the model input is assumed to occur uniformly in x over the right half of the model and B(x) is approximated to a step function.

ABSMUTE VALUE OF U AT COAST - - - LAND HEATED ANO m METERS ABOVE GRWND -.-.--.LAND COOLED

D l I I I I I I '

Figure 4. u velocity (cmls) at the coastline.

1 2 3 1 5 6 7

TIME IHou151

THE LAND-BREEZE SEA-BREEZE CIRCULATION 533

20

a -

16

3 - . iuv 2 3 12

r’ 10

8 -

Horizontal averages are taken over the model domain of length 2L. The N*2 at the beginning is I x I O - ~ S - ~ . u YELOClll U 5 K L m P S UGH

----u L A W -TED. Urn ,w*m - - - U LANO Cam. U K Y (NEI

-

-

-

-

Figure 5. u velocities (cm/s) at 45 glometres from the mast and 1.25 kilometres above the ground ----- land heated, 45km inland - - - - - land cooled, 45km over the ocean.

The available potential energy of the land-breeze will be discussed fist. The heat input is a linearly decreasing function of z. The height that the cooling finishes at is 1-25km with u at the ground 123cm s-’. After the heat has been removed N*2 of the layer below 1.25km is 1.5 x This can easily be seen by examining Fig. 8.

The deviation of u* from this new N*’ on the ocean side is now

6-25 z o*’(x,z) = -6.25 + for x < 0.

1-25 x 105’ The deviation from the mean over the land is

6.252 o*‘(x,z) = 6.25 - for x > 0.

1-25 x 105 A N N 1 OF lNum - WTFLW AT COAST

z2 2

6- 2 3 1 5 6 7 0

TME (HOURS)

Figure 6. Amount of inflow-outflow at the coast ( x 1O6cm2s-l) ----- land heated - * - land c001ed.

534 R. A. PEARSON

-L

-L This is consistent with the definition of N*, i.e. us’ (x,z) dx = 0. This results in

for both x > 0 and x < 0. Averaging over x and integrating between z = 0 and z = h = 1.25 x lo5 gives

(7) ‘ TOULKWETC ENEROY IN. - - -LAND KATED THEYODEL BOD( - a - - -LAND CWLED

Figure TYE IHWRSl

7. The total kinetic energy ( x 106cmZs-’) inside the model ----- land heated - . - . - land cooled.

box

Above 13km u* is zero and there is no contribution to the available potential energy. The total available potential energy (for the length 2L) in the land-breeze case is from Eq. (5 )

(6*25)’(1*25 x 10’) A.P.E. = 2L3 cm4 s-’

1.5 x 10-4

= 2.17 x 10” Lcm4 s-” (8) ORiGWL AND WER OCEAN I /

LAND COOLED

AND WER OCEAN

LAND HEATED

.AND

Figure 8. The new and average temperature profiles of the land-breeze (below) and sea-breeze (above).

THE LAND-BREEZE SEA-BREEZE CIRCULATION 535

The available potential energy in the heated land case is now calculated in a similar manner. The air above the land has a buoyancy of - 12.5cm at the ground decreasing linearly to 1.25km. N*2 after this heating is now 5 x lo-'. This can easily be seen by examining Fig. 8. For the sea breeze this gives

total A.P.E. = 6-41 x 10" Lcm4 s - ~ . (9)

Comparison between Eqs. (8) and (9) shows that for the experiments being considered, the fluid which has been heated has three times the available energy of the same fluid which is cooled, where the cooling is equal to the heating but opposite in sign. As the available energy is less, the strength and intensity of the circulation will be expected to be less. This is consistent with the results of the non-linear model.

In general the total available potential energy, for a given temperature change, is less for a stably stratified fluid when the fluid is cooled (the mean Brunt-Vaisala frequency increases), than when it is heated.

5. SUMMARY

This comparison shows that the sea-breeze would be expected to be stronger than the land-breeze even if the heat transfer were the same in the two cases. This difference in intensity between the land- and sea-breezes will be increased if there is less of a temperature contrast between the land and sea as was suggested by Defant (1951) but the weaker intensity does not rely on that mechanism alone.

ACKNOWLEDGMENTS

I wish to thank the National Center for Atmospheric Research which is supported by the National Science Foundation (U.S.A.) for the use of their CDC 6600 and 7600 Computers. This work was supported in part by the National Science Foundation under contract GA26452 at the University of Miami.

I appreciate the comments of Dr. C. Rooth, Dr. J. E. Geisler, and Professor H. Charnock.

Defant, F.

Estoque, M. A.

Green, J. S. A.

Lorenz, E. N.

McPherson, R. D.

1951

1960

1962

1970

1955

1967

1968

1970

1971 Neumann, J. and Mahrer, Y.

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Met. SOC.. 87, pp. 136-146. 'The sea breeze as a function of the prevailing synoptic

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circulation,' Quart. J. R. Met. Sor., 96, pp. 157-185. ' Available potential energy and the maintenance of the

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atmosphere, World Meteorological Organisation. ' A three-dimensional numerical study of the Texas coast

sea breeze,' Ph.D. Dissertation. University of Texas, Austin, U.S.A.

' A numerical study of the effect of a coastal irregularity on the sea breeze,' J. Appl. Met., 9, pp. 767-777.

' A theoretical study of the land and sea breeze circulation,' J. Atmos. Sci., 28, pp. 532-542.

536

Pearce, R. P.

Pearson. R. A.

R. A. PEARSON

1955 ' The calculation of a sea breeze circulation in terms of the differential heating across the coastline,' Quarr. J. R. Met. SOC., 81, pp. 251-281.

1973 ' Some aspects of the sea breeze front as shown by a nume- rical model,' J. Arms. Sci., 31, pp. 1050-1060.

1974 ' Consistent boundary conditions for numerical models of systems that admit dispersive waves,' Ibid., 32, pp. 1481-1489.